In this paper , we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the K-theoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration, this theorem reduces to the Atiyah-Singer index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured foliations ([9], Theorem, p. 136) and unlike the latter makes sense for arbitrary foliations, not necessarily gifted with a holonomy invariant transverse measure. The index in the Atiyah Singer theorem for families ([2]) is an element of the K-theory K(B) of the base space of the fibration. In the case of foliations the base B is the space of leaves of the foliation (V, F ). This space of leaves, as a topological space, is often degenerate (if the foliation is minimal there are no nontrivial open sets in V/F ). The algebra C(B) of continuous functions on B is replaced by a canonically defined C-algebra: C(V, F ), cf. [9], [10]. The K-theory K0(C (V, F )) of this C-algebra plays the role of K(B). In the case of a fibration C(V, F ) is (Morita) equivalent to C(B) so that K0(C (V, F )) = K(B). Let D be an elliptic differential operator along the leaves of the foliation (V, F ). Since D is elliptic it has an inverse modulo C(V, F ) hence it gives an element Inda(D) of K0(C (V, F )). Let us now describe the topological index. Let i be an auxiliary imbedding of the manifold V in R. Let N be the total space of the normal bundle to the leaves: Nx = (i∗(Fx)) ⊥ ⊂ R. Let us foliate Ṽ = V ×R by F , F(x,t) = Fx × {0}, so that the leaves of (Ṽ , F ) are just L = L × {t}, where L is a leaf of (V, F ) and t ∈ R. The map (x, ξ) → (x, i(x) + ξ) turns an open neighborhood of the 0-section in N into an open transversal T of the foliation (Ṽ , F ). For a suitable open neighborhood Ω of T in Ṽ , the C-algebra C(Ω, F ) of the restriction of F to Ω is (Morita) equivalent to C0(T ), hence the inclusion C (Ω, F ) ⊂ C(Ṽ , F ) yields a K-theory map: K(N) → K0(C (Ṽ , F )). Since C(Ṽ , F ) = C(V, F )⊗ C0(R ), one has, by Bott periodicity, the equality K0(C (Ṽ , F )) = K0(C (V, F )). Using the Thom isomorphism K(F ) is identified with K(N) so that one gets by the above construction, the topological index: Indt : K (F ) → K0(C (V, F )) . Our main result is the equality: Inda(D) = Indt([σD]) where σD is the longitudinal symbol of D and [σD] is its class in K(F ). In the first section, we formalize the elliptic pseudo-differential calculus for families of operators onX indexed by Y , in terms of the bivariant Kasparov theory. This gives a map of the K-theory with compact support K(T X × Y ) to the bivariant group KK(X,Y ). We then compute directly the Kasparov product of two such elements. In the second section we first recall the definition of [10] of the analytical element f ! ∈ KK(X,Y ) corresponding to a K-oriented map f from X to Y . We then prove that (idX)! is the unit of the ring KK(X,X). Using the computation of Section 1, we then prove the equality: (f ◦ g)! = g!⊗ f !. Computing f ! in the case

background

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=20&iss=6&rank=3

Feuilletages sur les variétés ouvertes

The Longitudinal Index Theorem for Foliations

Lectures on characteristic classes and foliations

Expanding attractors

/pdf/foliations-and-groups-of-diffeomorphisms-8ti01j3mau.pdf

Foliations and groups of diffeomorphisms

Homotopy and integrability

Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes

Submersions of open manifolds

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Abstract homotopy theory

Let M be a smooth manifold with tangent bundle TM. A k-plane field (or k-distribution) on M is a k-dimensional subbundle a of TM. Equivalently let a denote the section of the Grassmann bundle Gk(M) of k-planes associated to TM whose value at x e M is the k-plane ax c TM,. Two k-plane fields are homotopic if they are homotopic as sections of Gk(M). Homotopic k-plane fields are equivalent as k-plane bundles over M, but not conversely. If L is an injectively immersed, smooth submanifold of M such that TLX = ax c TMx for all x e L, L is called an integral submanifold of a. A kplane field a is called completely integrable if the following three equivalent conditions are satisfied. A. M is covered by open sets U with local coordinates x1, * , x" such that the submanifolds defined by xk+l = constant, * * *, xm = constant are integral submanifolds of a. B. a is smooth and through every point x e M there is an integral submanifold L of a. C. a is smooth and if X and Y are vector fields on M with Xx, Yx C ax for all x e M then the bracket [X, Y]x e ax. The equivalence of these conditions is the Frobenius theorem. An integrable k-plane field is also called a foliation (this is equivalent to other definitions) and the maximal connected integral submanifolds are called leaves. The leaves of a foliation partition the manifold. The existence theorem for ordinary differential equations says that smooth line-fields are always integrable. In general for k > 1 the set of kplane fields which are not integrable is open and dense in the space of sections of Gk(M). G. Reeb [16] has asked if the existence of a k-plane field on a manifold implies the existence of a foliation. He has given an example of a foliation of codimension one on S3. W. B. R. Lickorish [10] and, independently, S. P. Novikov and H. Zeischang have exhibited foliations of codimension one on any closed, orientable 3-manifold. In ? 4 we consider the unorientable

Foliations on 3-manifolds

/pdf/smooth-maps-transverse-to-a-foliation-3ostfgki5a.pdf

Feuilletages sur les variétés ouvertes

Citations

The Longitudinal Index Theorem for Foliations

Lectures on characteristic classes and foliations

Expanding attractors

Foliations and groups of diffeomorphisms

Homotopy and integrability

References

Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes

Submersions of open manifolds

Abstract homotopy theory

Foliations on 3-manifolds

Smooth maps transverse to a foliation

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Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes